Regular Maps in the Torus

This page links to three pages (links above) showing some of the regular
maps that can be drawn on the genus-1 orientable manifold, the torus. All
those with 50 or fewer faces, and their duals, are shown. For the purpose
of these pages, a "regular map" is defined here.

For other oriented manifolds, the number of such figures is small, but for
the torus, it is infinite. A reason why there are so many for the torus, and
a finite number for every other oriented 2-manifold, is that the torus has
an Euler characteristic of 0. Thus, once we have found one regular map, we
can stitch together several copies of it, to form another which still fits
on a torus.

As the "curvature" of the torus is 0, its vertices must be "flat": if they
are also fully symmetrical, they must be formed from four squares, or three
hexagons, or six triangles. Infinitely many regular maps of each of these
three types exist.

There is one regular map with four squares meeting at each vertex for each pair of
non-negative integers a,b (except for 0,0). Each has a number of faces equal to
a2+b2. Regular maps for integer pairs a,b with a<b exist,
but are not shown here; they are the enantiomorphs of those for b,a.

ARM disallows (in our notation) {4,4}(1,0), in which the single square
shares two edges with itself; and {4,4}(1,1), in which each of the two
squares shares each of its vertices (but no edge) with itself.

Any {4,4} can be cantellated, yielding a {4,4} with twice as many
vertices, faces and edges.

The regular maps with three hexagons meeting at each vertex are more complicated. They can
all be generated from pairs of number of the form a,b where a and b are either
both odd or both even. The number of faces of these regular maps is given by
(a2+3*b2)/4.

More than one such pair can generate the same regular regular map, for example {6,3}(2,4),
{6,3}(5,3), and {6,3}(7,1) are all the same regular map, with 13 faces. I have
arbitrarily chosen to list them in ascending order of the first parameter, which is necessarily also
descending order of the second parameter.

Most of these regular map are chiral, and so occur as enantiomorphic pairs. Only one (arbitrarily
chosen) member of each such pair is shown.

The notation {6,3}(a,b) used here is not consistent with
that used in ARM, page 19. Where ARM writes {6,3}(s,0) we write
{6,3}(s,s), and where ARM writes {6,3}(s,s) we write {6,3}(0, 2s).

ARM disallows regular maps which (in our notation) are not of either of the forms
{6,3}(s,s) and {6,3}(0, 2s), because they lack "full reflexional symmetry",
i.e. they are chiral. It also disallows (in our notation) {6,3}(1,1),
in which the single hexagon shares three edges with itself.

The regular maps with six triangles meeting at each vertex are the duals of those with three
hexagons. As for {6,3}(a,b), a and b must be either both odd or both even.
The number of faces of these polyhedra is given by (a2+3*b2)/2.

The notation used here {3,6}(a,b) is not consistent with that used in
ARM, and is as described above for {6,3}(a,b). Thus our
{6,3}(a,b) is the dual of our {3,6}(a,b).

ARM also disallows (in our notation) {3,6}(1,1), in which each of the two triangles
shares each of its vertices (but no edge) with itself.